 Critical points on \(\mathbb {F}_{n}(E)\)  of the map \({\Vert {{{q}}}\Vert }_M^{2\alpha } U({{q}})^2\)  (or \({\Vert {{{q}}}\Vert }_M^\alpha U({{q}})\) ) [1]}.
In all these formulations, central configurations appear as \(O(d)\) -orbits in \(\mathbb {F}_{n}(E)\) , where the action of the orthogonal group \(O(d)\)  on \(\mathbb {F}_{n}(E)\)  is diagonal \(g\cdot {{q}}= (g{{q}}_1,\ldots , g{{q}}_n)\) .
Define the space \(X\)  as
\(X = \left\lbrace  {{q}}\in \mathbb {F}_{n}(E) \colon \sum _{j=1}^n m_j {{q}}_j = {0}\right\rbrace .\) 
Central configurations and mutual differences
Let \({{n}}\)  be the set \({{n}}=\lbrace 1,2,\ldots , n\rbrace \)  and \(C^0\)  the vector space of all maps from \({{n}}\)  to \(E\) : \(C^0 = \lbrace  {{q}}\colon {{n}}\rightarrow E \rbrace \) . Let \(\mathbb {F}_{n}(E) \subset C^0\)  denote the inclusion sending \({{q}}\in \mathbb {F}_{n}(E)\)  to the map \({{q}}\colon {{n}}\rightarrow E\)  defined by \({{q}}(j) = {{q}}_j\)  for each \(j\in {{n}}\) .
Now, let \(\tilde{{{n}}}\)  denote the set of all \({n {2}}\)  subsets in \({{n}}\)  with two elements: \(\tilde{{{n}}}=\lbrace \lbrace 1,2\rbrace ,\lbrace 1,3\rbrace ,\ldots \) , \(\lbrace n-1,n\rbrace \rbrace \) . Let \(C^1\)  denote the vector space of all maps from \(\tilde{{{n}}}\)  to \(E\) :
\(C^1 = \lbrace  {{q}}\colon \tilde{{{n}}}\rightarrow E \rbrace .\) 
It is isomorphic to \(E^{\tilde{n}}\) , where \(\tilde{n}={n {2}}\) . Note that if \(E^{{{n}}^2}\)  denotes that vector space of all maps \({{q}}\colon {{n}}^2 \rightarrow E\) , where \({{n}}^2 = {{n}}\times {{n}}\)  (and hence if \({{q}}\in E^{{{n}}^2}\) , we can denote \({{q}}_{ij} = {{q}}( (i,j) ) \in E\) ), there is an embedding \(C^1 \subset E^{{{n}}^2}\) , by sending an element \({{q}}\in C^1\)  to the map \({{q}}^{\prime }\colon {{n}}^2 \rightarrow E\)  defined by
\({{q}}^{\prime }_{ij} ={\left\lbrace \begin{array}{ll}{{q}}(\lbrace i,j\rbrace ) & \text{if $i<j$}, \\-{{q}}(\lbrace i,j\rbrace ) & \text{if $i>j$}, \\0 & \text{if $i=j$.}\end{array}\right.}\) 
In fact, we are identifying elements in \(C^1\)  with the skew-symmetric elements in \(E^{{{n}}^2}\)  (that is, maps \({{q}}_{ij} + {{q}}_{ji} = {0}\in E\) ). Given \({{q}}\in C^1\)  with an abuse of notation we will write \({{q}}_{ij}\)  instead of \({{q}}^{\prime }_{ij}\) , and \(ij\)  instead \((i,j)\) .
If \(K\)  is an abstract simplicial complex, recall that the simplicial chain complex of \(K\)  with real coefficients, denoted by \(C_*(K;\mathbb {R})\) , is defined as follows: for each \(k\in \mathbb {Z}\) , the chain group \(C_k(K;\mathbb {R})\)  is the vector space of all the \(\mathbb {R}\) -linear combinations of \(k\) -dimensional simplexes of \(K\) ; the boundary homomorphism \(\partial _k \colon C_k(K;\mathbb {R}) \rightarrow C_{k-1}(K;\mathbb {R})\)  is defined as \(\partial _k(\sigma ) = \sum \limits _{j=0}^k (-1)^j \sigma d_j \)  for each \(k\) -simplex \(\sigma \)  of \(K\)  and 0 otherwise, where \(d_j\)  is the \(j\) -th face map. More precisely, all simplices in \(K\)  can be ordered, and elements in \(C_k(K;\mathbb {R})\)  will be linear combinations of ordered \(k\) -simplices in \(K\) . An ordered \(k\) -simplex with vertices \(x_0,\ldots , x_{k}\)  will be denoted either as \([x_0,\ldots , x_k]\)  or simply as \(x_0\ldots x_k\) . With this notation, the \(j\) -th face map sends \(\sigma = [x_0,\ldots ,x_j,\ldots , x_k]\)  to \(\sigma d_j = [ x_0,\ldots , \widehat{x}_j, \ldots , x_k]\) , where \(\widehat{x}_j\)  means that the \(j\) -th element is canceled.
By taking homomorphisms valued in an \(\mathbb {R}\) -vector space \(E\) , the chain complex \(C_k(K;\mathbb {R})\)  yields the simplicial cochain complex with coefficients in \(E\) : the cochain groups are defined as all the linear homomorsphisms \(C^k(K;E) = \hom _\mathbb {R}(C_k(K;\mathbb {R}), E)\) , and the coboundary homomorphisms \(\delta ^k \colon C^k(K;E) \rightarrow C^{k+1}(K;E)\)  are defined for each \(k\)  by
\(\delta ^k( \eta ) = \eta \partial _{k+1} \colon \ C_{k+1}(K;\mathbb {R}) \rightarrow E\) 
for each cochain \(\eta \colon C_{k}(K;\mathbb {R}) \rightarrow E\) . The kernel of \(\delta ^k\)  is the group of cocycles, and it is denoted as \(Z^k(K;E) = \ker \delta ^k \subset C^k(K;E)\) .
Now, let \(\Delta ^{n-1}\)  denote the standard (abstract) simplex with \(n\)  vertices \(\lbrace 1,2,\ldots , n\rbrace \) . Then the vector spaces \(C^k\)  defined above for \(k=0,1\)  are exactly the groups of \(k\) -dimensional simplicial cochains (with coefficients in the vector space \(E\) ) of the simplicial complex \(\Delta ^{n-1}\) : \(C^0 = C^0(\Delta ^{n-1};E)\)  and \(C^1 = C^1(\Delta ^{n-1};E)\) . A 0-simplex of \(\Delta ^{n-1}\)  is simply an element \(j \in {{n}}\) , and hence a 0-dimensional cochain is an \(n\) -tuple \({{q}}_j\) , i.e., a map \({{q}}\colon {{n}}\rightarrow E\) . Furthermore, a 1-dimensional cochain is a map \({{q}}\)  defined with values in \(E\)  and as domain the set of 1-dimensional simplices of \(\Delta ^{n-1}\) , i.e., pairs \(ij\)  with \(1\le i<j\le n\) .
In simpler terms, for each \(i,j\)  such that \(1\le i < j \le n\) , let \({{q}}_{ij} \in E\)  denote the \(ij\) -the component of a vector in \(E^{\tilde{n}}\) , and for \(i> j\) , the variable \({{q}}_{ij}\)  is defined by the property that \(\forall \, i,j\) , \({{q}}_{ij} + {{q}}_{ji} = {0}\) .
The coboundary operator \(\delta ^0 \colon C^0 \rightarrow C^1\)  is the map defined by \(\delta ^0 {{q}}= {{q}}\partial _1\)  for each \({{q}}\in C^0\) , and hence
\(\delta ^0({{q}}) ({i,j}) = {{q}}_j - {{q}}_i \in E\) 
for all \(i\) , \(j\) . In fact, for each \({{q}}\colon {{n}}\rightarrow E\) , \(\delta ^0({{q}}) (ij) = {{q}}\partial _1 [i,j] = {{q}}( j-i) = {{q}}_j - {{q}}_i\) . For \(k=1\) , the coboundary operator is defined as \(\delta ^1\colon C^1 \rightarrow C^2\)  as
\(\delta ^1({{q}})(ijk) = {{q}}\partial _2(ijk) = {{q}}( jk - ik + ij) = {{q}}_{ij} + {{q}}_{jk} + {{q}}_{ki}.\) 
Moreover, since the simplex \(\Delta ^{n-1}\)  is contractible, its cohomology groups are trivial except for \(k-0\) , and therefore for each \(k \ge 0\)
\(Z^{k+1}\big (\Delta ^{n-1};E\big ) = \ker \delta ^{k+1} = \operatorname{Im}\delta ^{k}.\) 
With an abuse of notation, when not necessary the subscript of \(\partial _k\)  and the supscript in \(\delta ^k\)  will be omitted.
For each \({{q}}\in C^1\) , let \({{Q}}\)  be defined as \({{Q}}_{jk} = \frac{{{q}}_{jk}}{\vert {{{q}}_{jk}}\vert ^{\alpha +2}}\) . Note that \({{Q}}_{jk} = \Psi _\gamma ({{q}}_{jk})\) 
with \(\gamma =\alpha +2\)  where \(\Psi _\gamma ({{x}}) = \frac{{{x}}}{\vert {{{x}}}\vert ^\gamma }\)  for each \({{x}}\in E\) . It turns out that the map \(\Psi _\gamma \colon E\setminus \lbrace {0}\rbrace  \rightarrow E \setminus \lbrace {0}\rbrace \)  is a diffeomorphism with inverse \(\Psi _{\hat{\gamma }}\)  where \(\hat{\gamma }= \frac{\gamma }{\gamma -1}\) .
Consider now that if one defines \({{q}}_{ij}={{q}}_i - {{q}}_j\) , one can read equation (REF ) as (as a consequence of equation (REF ))
\(\lambda {{q}}_j=-\alpha \sum _{k\ne j} m_k\frac{{{q}}_{jk}}{\vert {{{q}}_{jk}}\vert ^{\alpha +2}}=-\alpha \sum _{k\ne j} m_k {{Q}}_{jk},\) 
and hence

- qij  =ki mk Qik-kj mk Qjk =
k{i,j} mk (Qik + Qkj)+ mj Qij + mi Qij
 =k{i,j} mk (Qik + Qkj + Qji)-k{i,j} mk Qji+ mj Qij + mi Qij
=k{i,j} mk (Qik + Qkj + Qji)+(k mk) Qij

- qij = k{i,j} mk (Qik + Qkj + Qji)+Qij.
Proposition 2.1 
The \((\) linear\()\)  map \(P_m \colon C^1 \rightarrow C^1\)  defined by
\(\left( P_m({{Q}}) \right)_{ij} = \sum _{k\notin \lbrace i,j\rbrace } m_k ({{Q}}_{ik} + {{Q}}_{kj} + {{Q}}_{ji})+{{Q}}_{ij}\) 
is a projection from \(C^1\)  onto \(Z^1\subset C^1\) , where \(Z^1=\ker \delta ^1 \colon C^1 \rightarrow C^2\)  is the subspace of 1-cocycles.

Consider the homomorphism \(\pi _m\)  defined on the vector space of simplicial 1-chains \(C_1(\Delta ^{n-1},\mathbb {R})\)  with real coefficient, as
\(\pi _m([i,j]) = [i,j] - \sum _{k\ne \lbrace i,j\rbrace } m_k \partial _2([i,j,k]),\) 
where \(\partial _2 \colon C_2 \rightarrow C_1\)  is the boundary homomorphism in dimension 2. Then for any \({{Q}}\in C^1\)  and any \(i\) , \(j\)  with \(i\ne j\)
\(P_m({{Q}})[i,j] = {{Q}}( \pi _m [i,j]) .\) 
For each 2-simplex \([a,b,c]\)  of \(\Delta ^{n-1}\)  one has \(\partial _2 ([a,b,k] + [b,c,k] + [c,a,k]) = \partial _2 [a,b,c]\)  for each \(k\ne \lbrace a,b,c\rbrace \) , and hence
\(\pi _m \partial _2 [a,b,c] = \pi _m([a,b] + [b,c] + [c,a]) =[a,b] - \sum _{k\ne \lbrace a,b\rbrace } m_k \partial _2 [a,b,k]\\\hphantom{\pi _m \partial _2 [a,b,c] =}{} +[b,c] - \sum _{k\ne \lbrace b,c\rbrace } m_k \partial _2 [b,c,k] + [c,a] - \sum _{k\ne \lbrace c,a\rbrace } m_k \partial _2 [c,a,k]\\\hphantom{\pi _m \partial _2 [a,b,c]}{} =\partial _2 [a,b,c]- \sum _{k\notin \lbrace a,b,c\rbrace } m_k \partial _2 [a,b,c]-m_c \partial _2 [a,b,c] -m_a \partial _2 [b,c,a] -m_b \partial _2 [c,a,b]\\\hphantom{\pi _m \partial _2 [a,b,c]}{} = \partial _2 [a,b,c] - \left( \sum _k m_k \right) \partial _2 [a,b,c] = 0\\\Rightarrow \pi _m \partial _2 = 0.\) 
As a consequence, \(\pi _m\)  is a projection, since for each \(i\) , \(j\) , \(i\ne j\) ,
\(\pi _m^2[i,j] =\pi _m\left([i,j] - \sum _{k\notin \lbrace i,j\rbrace } m_k \partial _2([i,j,k])\right) = \pi _m[i,j] - \sum _{k\notin \lbrace i,j\rbrace } m_k \pi _m\partial _2[i,j,k] =\pi _m[i,j].\) 
Therefore, also \(P_m\colon C^1 \rightarrow C^1\)  is a projection
\(P^2_m ( {{Q}}) [i,j] = P_m ( {{Q}}) (\pi _m [i,j] ) = {{Q}}( \pi _m^2 [i,j] ) = {{Q}}( \pi _m [i,j] ) = P_m({{Q}})[i,j].\) 
Moreover, since
\(\partial _1 \pi _m = \partial _1\) 
it follows that the projection \(P_m\)  is onto the subspace of all 1-cocycles in \(C^1\) , denoted in short by \(Z^1\) . In fact, since \(\pi _m \partial _2 =0\) ,
\(\delta ^1 P_m {{Q}}= P_m {{Q}}\partial _2 = {{Q}}\pi _m \partial _2 = 0 \Rightarrow \operatorname{Im}(P_m) \subset Z^1\) 
and, by (REF ), for each cocycle \({{z}}\in Z^1 \iff {{z}}= \delta ^0 {{x}}\)  one has
\(P_m {{z}}= P_m \delta ^0 {{x}}= P_m {{x}}\partial _1 = {{x}}\partial _1 \pi _m = {{x}}\partial _1 = \delta ^0 {{x}}= {{z}}\) 
and hence \(\operatorname{Im}(P_m) \supset Z^1\) . We can conclude, as claimed, that \(\operatorname{Im}(P_m) = Z^1\) .
As examples, for \(d=1\)  and \(n=3,4\)  the matrices of the projection \(P_m\)  are
\(\begin{bmatrix}m_{1}+m_{2}& m_{3} & -m_{3} \\m_{2} &m_{1}+m_{3}& m_{2} \\-m_{1} & m_{1} &m_{2}+m_{3}\end{bmatrix},\\\begin{bmatrix}m_{1}+m_{2}& m_{3} & m_{4} & -m_{3} & -m_{4} &0 \\m_{2} &m_{1}+m_{3}& m_{4} & m_{2} &0& -m_{4} \\m_{2} & m_{3} &m_{1}+m_{4}&0& m_{2} & m_{3} \\-m_{1} & m_{1} &0&m_{2}+m_{3}& m_{4} & -m_{4} \\-m_{1} &0& m_{1} & m_{3} &m_{2}+m_{4}& m_{3} \\0& -m_{1} & m_{1} & -m_{2} & m_{2} &m_{3}+m_{4}\end{bmatrix}.\) 
In fact, for \(n=3\)  the space of cochains \(C^1\)  has standard coordinates \({{Q}}_{ij}\)  for \(ij\in \lbrace 12,13,23\rbrace \) , and by Proposition REF  the projection \(P_m\)  in these coordinates is defined by
\(\left( P_m({{Q}}) \right)_{12} =\sum _{k\notin \lbrace 1,2\rbrace } m_k ({{Q}}_{1k} + {{Q}}_{k2} + {{Q}}_{21})+{{Q}}_{12} =m_3 ( {{Q}}_{13} + {{Q}}_{32} + {{Q}}_{21}) +{{Q}}_{12},\\\left( P_m({{Q}}) \right)_{13} =\sum _{k\notin \lbrace 1,3\rbrace } m_k ({{Q}}_{1k} + {{Q}}_{k3} + {{Q}}_{31})+{{Q}}_{13} =m_2({{Q}}_{12} + {{Q}}_{23} + {{Q}}_{31}) + {{Q}}_{13},\\\left( P_m({{Q}}) \right)_{23} =\sum _{k\notin \lbrace 2,3\rbrace } m_k ({{Q}}_{2k} + {{Q}}_{k3} + {{Q}}_{32})+{{Q}}_{23} =m_1( {{Q}}_{21}+{{Q}}_{13} + {{Q}}_{32} ) + {{Q}}_{23},\) 
from which it follows that
\(\left( P_m({{Q}}) \right)_{12} =(1-m_3) {{Q}}_{12}+ m_3 {{Q}}_{13}- m_3 {{Q}}_{23},\\\left( P_m({{Q}}) \right)_{13} =m_2{{Q}}_{12}+ (1-m_2){{Q}}_{13}+ m_2 {{Q}}_{23},\\\left( P_m({{Q}}) \right)_{23} =- m_1{{Q}}_{12}+ m_1 {{Q}}_{13}+ (1-m_1){{Q}}_{23}.\) 
The same argument yields the matrix for \(n=4\) , in coordinates \({{Q}}_{ij}\)  for \(ij\)  in the order 12, 13, 14, 23, 24, 34.
Consider the following scalar product on \(C^1\) , similar to the mass-metric on the configuration space: for \({{v}},{{w}}\in C^1\) ,
\({\langle {{{v}}},{{{w}}}\rangle }_{M} =\sum _{i<j} m_i m_j \left({{v}}_{ij} \cdot {{w}}_{ij}\right),\) 
where as above the dot denotes the standard \(d\) -dimensional scalar product in \(E\) . It is the mass-metric on \(C^1\) , and as above \({\Vert {{{v}}}\Vert }_M^2 = {\langle {{{v}}},{{{v}}}\rangle }_{M}\) . It follows that
\({\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M} =\sum _{i<j}m_i m_j \left( {{v}}_{ij} \cdot \left( P_m({{w}}) \right)_{ij} \right)\\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{} =\sum _{i<j} m_i m_j \left( {{v}}_{ij} \cdot \left(\sum _{k\notin \lbrace i,j\rbrace } m_k ({{w}}_{ik} + {{w}}_{kj} + {{w}}_{ji})+{{w}}_{ij} \right) \right) \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{} = \sum _{i<k} \sum _{k\notin \lbrace i,j\rbrace }m_i m_j m_k ( {{v}}_{ij} \cdot ({{w}}_{ik}+{{w}}_{kj} + {{w}}_{ji}) )+\sum _{i<j} m_i m_j {{v}}_{ij} \cdot {{w}}_{ij} \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{} = \sum _{a<b<c} m_a m_b m_c\big ({{v}}_{ab}\cdot ( {{w}}_{ac} + {{w}}_{cb} + {{w}}_{ba}) \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}=}{}+ {{v}}_{ac} \cdot ( {{w}}_{ab} + {{w}}_{bc} + {{w}}_{ca})+{{v}}_{bc} \cdot ( {{w}}_{ba} + {{w}}_{ac} + {{w}}_{cb})\big )+\sum _{i<j} m_i m_j {{v}}_{ij} \cdot {{w}}_{ij} \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{}= - \sum _{a<b<c} m_a m_b m_c\left(({{v}}_{ab}+{{v}}_{bc}+{{v}}_{ca})\cdot ( {{w}}_{ab} + {{w}}_{bc} + {{w}}_{ca})\right)\\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}=}{}+ \sum _{i<j} m_i m_j {{v}}_{ij} \cdot {{w}}_{ij} \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{}=\sum _{i<k} \sum _{k\notin \lbrace i,j\rbrace }m_i m_j m_k ( ({{v}}_{ik}+{{v}}_{kj} + {{v}}_{ji}) \cdot {{w}}_{ij} )+\sum _{i<j} m_i m_j {{v}}_{ij} \cdot {{w}}_{ij} \\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{} =\sum _{i<j} m_i m_j \left( \left(\sum _{k\notin \lbrace i,j\rbrace } m_k ({{v}}_{ik} + {{v}}_{kj} + {{v}}_{ji})+{{v}}_{ij} \right) \cdot {{w}}_{ij} \right)\\\hphantom{{\langle { {{v}}},{ P_m ({{w}}) }\rangle }_{M}}{}= {\langle { P_m ({{v}})},{ {{w}}}\rangle }_{M},\) 
hence the following proposition holds.
Proposition 2.2 The projection \(P_m\colon C^1 \rightarrow Z^1\subset C^1\)  is orthogonal \((\) self-adjoint\()\)  with respect to the scalar product \({\langle {-},{-}\rangle }_{M}\)  in (REF ) defined on \(C^1\) .

Now, consider the subspace \(X\subset \mathbb {F}_{n}(E)\)  of all configurations with center of mass in 0: \(X=\lbrace  {{q}}\in \mathbb {F}_{n}(E) \colon \sum _j m_j {{q}}_j = {0}\rbrace \) , i.e., of all \({{q}}\in C^0\)  such that \({{q}}\sum _j m_j [j] = {0}\) . The coboundary morphism \(\delta ^0_{|X}\colon X \subset C^0 \rightarrow C^1\)  induces an isomorphism \(\delta ^0_{|X} \colon X \rightarrow Z^1\) . Moreover, since if \({{q}}\in X\)  then
\(2 \sum _{i<j} m_im_j \vert {{{q}}_i -{{q}}_j}\vert ^2 =\sum _{i,j} m_i m_j \vert {{{q}}_i -{{q}}_j}\vert ^2 = \sum _{i,j} m_i m_j \left(\vert {{{q}}_i}\vert ^2 - 2{{q}}_i \cdot {{q}}_j + \vert {{{q}}_j}\vert ^2\right) \\\hphantom{2 \sum _{i<j} m_im_j \vert {{{q}}_i -{{q}}_j}\vert ^2}{} =2 \sum _{i} m_i \vert {{{q}}_i}\vert ^2 - 2 \left( \sum _{j} m_j {{q}}_j \right)^2 =2 \sum _{i} m_i \vert {{{q}}_i}\vert ^2\) 
the isomorphism \(\delta ^0_{|X}\colon X \rightarrow Z^1\)  is an isometry, where \(X\)  and \(Z^1\)  have the mass-metrics. Explicitly, for each \({{q}}\in X\) ,
\({\Vert {{{q}}}\Vert }_M = {\Vert { \delta ^0 ({{q}})}\Vert }_M,\) 
where the two norms with the same symbol, with an abuse of notation, are actually different norms in \(C^0\)  and \(C^1\)  respectively.
Furthermore, the potential \(U\)  is the composition of the restriction to \(X\)  of the coboundary map \(\delta ^0\)  with the map \(\tilde{U}\colon C^1 \rightarrow \mathbb {R}\)  (partially) defined by
\(\tilde{U}({{q}}) =\sum _{i<j} m_im_j \vert {{{q}}_{ij}}\vert ^{-\alpha },\) 
as illustrated in the following diagram
\(@C+24pt@R+24pt{C^0 [r]^{\delta ^0} & C^1 [r]^{\tilde{U}} @/_1em/[d]_{P_m} & \mathbb {R}\\X @{ >->}[u] [r]_{\delta ^0_{|X} }^{\cong } & Z^1. @{ >->}@/_1em/[u]}\) 
Now, recall that (condition (CC4)) a configuration \({{q}}\in \mathbb {F}_{n}(E) \subset C^0\)  is a central configuration is and only if it is a critical points of the map \({\Vert {{{q}}}\Vert }_M^2+ U({{q}})\) , defined on \(\mathbb {F}_{n}(E)\) . It is easy to see that this is equivalent to say that \({{q}}\)  is a critical point of the map \({\Vert {{{q}}}\Vert }_M^2+ U({{q}})\)  restricted to \(X\) . But this means that \(\delta ^0_{|X}\)  sends central configurations in \(X\)  to all the critical points of the map \({\Vert {\tilde{{{q}}}}\Vert }_M^2+ \tilde{U}(\tilde{{{q}}})\)  (defined on \(C^1\) ) restricted to the space of 1-cocycles \(Z^1\) .
Hence, the following theorem holds.
Theorem 2.3  Central configurations are critical points of the function partially defined as \(C^1 \rightarrow \mathbb {R}\)
\({{q}}\mapsto \sum _{i<j} m_im_j\left( \vert {{{q}}_{ij}}\vert ^{-\alpha } + \vert {{{q}}_{ij}}\vert ^2 \right)\) 
restricted to the space of 1-cocycles \(Z^1\subset C^1\) .
A co-chain \({{q}}\in C^1\)  is a central configuration if and only if there exists \(\lambda \in \mathbb {R}\)  such that \(\lambda {{q}}= P_m( \Psi ({{Q}}))\) , where \({{Q}}_{ij} = \frac{{{q}}_{ij}}{|{{q}}_{ij}|^{\alpha +2}}\)  for each \(i\) , \(j\)  and \(P_m \colon C^1 \rightarrow Z^1\subset C^1\)  is the orthogonal projection defined in Proposition REF , which sends \(C^1\)  onto the space of 1-cocycles.

Remark 2.4 Since the function \(r^{-\alpha } + r^2\)  is convex on \((0,\infty )\) , Theorem REF  implies that the restriction of \(\tilde{U}\)  to each component of \(Z^1\)  minus collisions is convex for \(d=1\) , and hence one derives the existence (and uniqueness) of Moulton collinear central configurations.


Hessians and indices
Let \(P\in \mathbb {F}_{n}(E)\)  be a central configuration, with mass-norm \(r={\Vert {P}\Vert }_M\) . As in the case \(r=1\) , seen in (CC1), it is a critical point of the restriction of the potential function \(U\)  to the inertia ellipsoid \(S= \lbrace  {{q}}\in \mathbb {F}_{n}(E) \colon {\Vert {{{q}}}\Vert }_M = r \rbrace \) . As such, its Morse index is the Hessian of the restriction \(U|_{S}\) , which is a bilinear form defined on the tangent space \(T_P S\) . The Hessian of \(f=U|_{S}\)  at a critical point \(P\in S\)  can be computed in general as \(D^2f(P)[{{u}}(P),{{v}}(P)] = ((D_{{u}}D_{{v}}f - D_{D_{{u}}{{v}}}) f) (P) = (D_{{u}}(D_{{v}}f))(P) \) , where \({{u}}\)  and \({{v}}\)  are vector fields on \(S\)  (see, e.g., [2]}). This yields the well-known formula of the Hessian in terms of second derivatives with respect to a local chart (see also [3]}). Given the mass-metric, the Hessian can be written as \(D^2f[{{u}},{{v}}] = {\langle { D_{{u}}\hat{\nabla }_Mf },{ {{v}}}\rangle }_{M}\)  or as the self-adjoint endomorphism \(T_PS \rightarrow T_PS\)  defined by \({{u}}\in T_PS \mapsto D_{{u}}\hat{\nabla }_Mf(P) \in T_PS\) , where \(\hat{\nabla }_M\)  denotes the gradient induced by the mass-metric restricted to \(S\)  (hence \(\hat{\nabla }_Mf (P)\)  is the projection of \(\nabla _MU(P)\)  to \(T_PS\) , orthogonal with respect to \({\langle {-},{-}\rangle }_{M}\) ).
If \({{N}}\)  denotes a vector normal to the tangent space \(T_PS\)  (such as \(P-O\) , where \(O\)  denotes the origin of the Euclidean space \(E\) ), the projection \(\hat{\nabla }_Mf(P)\)  is equal to \(\nabla _MU - \frac{{\langle {\nabla _MU},{{{N}}}\rangle }_{M}}{{\Vert {{{N}}}\Vert }_M^2} {{N}}\)  evaluated at \(P\) . If \({{N}}=P-O\) , by Euler formula \({\langle {\nabla _MU},{{{N}}}\rangle }_{M} = -\alpha U\) , and because \(P\)  is a critical point of \(f=U|_S\)  and \({{u}}\in T_PS\)  the derivative \(D_{{u}}\frac{\alpha U({{q}})}{{\Vert {{{q}}}\Vert }_M^2}\)  vanishes at \(P\) , and hence
\(D_{{u}}\hat{\nabla }_Mf(P) =D_{{u}}\left(\nabla _MU({{q}}) + \frac{\alpha U({{q}})}{{\Vert {{{q}}}\Vert }_M^2} {{N}}\right) (P) =D_{{u}}(\nabla _MU)(P)+ \frac{\alpha U(P)}{{\Vert {P}\Vert }_M^2}D_{{u}}({{q}}) \\\hphantom{D_{{u}}\hat{\nabla }_Mf(P)}{} = D_{{u}}\left(\nabla _MU-\lambda \nabla _M\frac{{\Vert {{{q}}}\Vert }_M^2}{2}\right),\) 
where \(\lambda \)  is as above the constant \(-\frac{\alpha U(P)}{{\Vert {P}\Vert }_M^2}\) . Hence the following lemma holds.
Lemma 3.1 
If \(P\)  is a critical point of the restriction \(U|_{S_r}\) , with the inertia ellipsoid \(S_r =\lbrace  {{q}}\in \mathbb {F}_{n}(E) \colon {\Vert {{{q}}}\Vert }_M = r \rbrace \)  and with \(\lambda \)  defined as \(\lambda = -\frac{\alpha U(P)}{{\Vert {P}\Vert }_M^2} = -\frac{\alpha U(P)}{r^2}\) , then \(P\)  is a critical point of the function \(U({{q}})-\frac{\lambda }{2} {\Vert {{{q}}}\Vert }_M^2\) ; moreover, the Hessian of \(U|_{S_r}\)  at \(P\)  is the restriction to the tangent space \(T_P{S_r}\)  of the Hessian of the map \(U({{q}})-\frac{\lambda }{2} {\Vert {{{q}}}\Vert }_M^2\)  defined on \(\mathbb {F}_{n}(E)\) , evaluated \(P\) .

Proposition 3.2 
If \(P\in \mathbb {F}_{n}(E)\)  is a central configuration, then the Morse index at \(P\)  of the restriction \(f=U|_{S_r}\)  is equal to the Morse index at \(P\)  of the function \(F({{q}}) = U({{q}})-\frac{\lambda }{2} {\Vert {{{q}}}\Vert }_M^2\) , where \(\lambda \)  and \(S_r\)  are as above. Furthermore, the direction parallel to \(P-O\)  is an eigenvector of the Hessian of \(F\) , with (positive) eigenvalue equal to \(-\lambda (\alpha +2)>0\) .

Since \(\nabla _M{U({{q}})}\)  is homogeneous of degree \(-\alpha -1\) , if \({{N}}= P-O\)  one has \(D_{{N}}( \nabla _M{U} )(P) = -(\alpha +1) \nabla _M{U(P)} = -\lambda (\alpha +1) {{N}}\) . Therefore
\(D_{{N}}\left(\nabla _MU-\lambda \nabla _M\frac{{\Vert {{{q}}}\Vert }_M^2}{2}\right) (P)=-\lambda (\alpha +1) {{N}}- \lambda {{N}}=-\lambda (\alpha +2) {{N}}.\qquad \mathrm {}\) 
Now, consider the function \(f\colon C^1 \rightarrow \mathbb {R}\)  defined on cochains in Theorem REF  as
\(f({{q}}) = \sum _{i<j} m_im_j\left( \vert {{{q}}_{ij}}\vert ^{-\alpha } + \vert {{{q}}_{ij}}\vert ^2 \right).\) 
The following proposition links its Hessian with the Hessian of the function \(F\)  of Proposition REF , for \(\lambda = -2\) .
Proposition 3.3 
Let \({{q}}\in \mathbb {F}_{n}(E)\)  be a central configuration which is a critical point of the function \(F({{q}}) = U({{q}}) + {\Vert {{{q}}}\Vert }_M^2\)  in \(C^0\)  \((\) and hence \({{q}}\in X)\) . Let \(H\)  be the Hessian of \(F\)  at \({{q}}\)  \((\) with respect to the mass-metric in \(C^0)\) , and \( \tilde{H}\)  the Hessian matrix of the composition \(f \circ P_m \)  at \(\delta ^0({{q}})\in Z^1 \subset C^1\)  \((\) with respect to the mass-metric in \(C^1)\) . Then the non-zero eigenvalues of \(\tilde{H}\)  are the same as the non-zero eigenvalues of \(H\) , except for the eigenvalue 2 occurring in \(H\)  with multiplicity \(\dim E\)  \((\) which corresponds to the group of translations in \(E\) , or equivalently the orthogonal complement of \(X\)  in \(C^0)\) .

Since \(U\)  is invariant with respect to translations in \(C^0\) , \(H\)  has the autospace \({{q}}_1={{q}}_2=\dots ={{q}}_n \subset C^0\)  (which is the tangent space of the group of translations acting on \(\mathbb {F}_{n}(E)\) , and is orthogonal to \(X\)  with respect to the mass-metric), over which \(D^2U\)  vanishes and \(D^2 {\Vert {{{q}}}\Vert }_M^2 = 2\) ; hence it is an eigenspace with eigenvalue 2. The rest of eigenvalues of \(H\)  correspond via the isometric embedding \(\delta ^0_{|X}\)  to eigenvalues of the restriction of \(f\)  to \(Z^1\) , and hence to the eigenvalues in \(Z^1\)  of the composition \(f \circ P_m\) . The orthogonal complement of \(Z^1\) , which is the kernel of \(P_m\) , yields zero eigenvalues to \(\tilde{H}\) .

Simple proofs of some corollaries
Equations () can be written as the following:
\(\frac{\lambda }{\alpha } {{q}}_{ij} +{{Q}}_{ij} =\sum _{k\notin \lbrace i,j\rbrace } m_k ({{Q}}_{ij} + {{Q}}_{jk} + {{Q}}_{ki} ).\) 
Now, consider for each triple \(i\) , \(j\) , \(k\)  the corresponding term \({{Q}}_{ijk} = {{Q}}_{ij} + {{Q}}_{jk} + {{Q}}_{ki}\) . We give some very simple proofs to some well-known propositions (actually generalizing them to any homogeneity \(\alpha \) ), that follow from the following simple geometric lemma.
Lemma 4.1 
Let \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  be three non-collinear points in \(E\) . Then \({{Q}}_{123} = {0}\)  if and only if \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  are vertices of an equilateral triangle.
Furthermore, there exists \(c\in \mathbb {R}\)  such that \({{Q}}_{123} = c {{q}}_{12}\)  if and only if \(\vert {{{q}}_{13}}\vert  = \vert {{{q}}_{23}}\vert \) , that is, if and only if the triangle with vertices in \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  is isosceles in \({{q}}_3\) .

If \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  are not collinear (in \(E\) ), then the differences \({{q}}_{12}\) , \({{q}}_{13}\)  and \({{q}}_{23}\)  generate a plane. Since \({{q}}_{12} + {{q}}_{23} + {{q}}_{31} = {0}\) , \({{Q}}_{123}={0}\)  implies
\({{Q}}_{123} = {{Q}}_{12} + {{Q}}_{23} + {{Q}}_{31} =\frac{{{q}}_{12}}{\vert {{{q}}_{12}}\vert ^{\alpha +2}} +\frac{{{q}}_{23}}{\vert {{{q}}_{23}}\vert ^{\alpha +2}} +\frac{{{q}}_{31}}{\vert {{{q}}_{31}}\vert ^{\alpha +2}}= {0}= {{q}}_{12} + {{q}}_{23} +{{q}}_{31} .\) 
By taking barycentric coordinates in the plane generated by the three points, it follows that \({{Q}}_{123} = {0}\)  if and only if \(\vert {{{q}}_{12}}\vert ^{\alpha +2} = \vert {{{q}}_{23}}\vert ^{\alpha +2} = \vert {{{q}}_{31}}\vert ^{\alpha +2}\) , that is, if and only if the three points are vertices of an equilateral triangle.
If \(c_1\) , \(c_2\)  and \(c_3\)  are three non-zero real numbers such that \(c_3 {{q}}_{12} + c_1 {{q}}_{23} + c_2 {{q}}_{31} = c {{q}}_{12}\) , then \((c_3-c) {{q}}_{12} + c_1 {{q}}_{23} + c_2 {{q}}_{31} = {0}\) , and as above this implies \(c_3-c=c_1=c_2\) . Hence if \({{Q}}_{123} = c {{q}}_{12}\) , it holds that \(\vert {{{q}}_{23}}\vert ^{\alpha +2} = \vert {{{q}}_{13}}\vert ^{\alpha +2} \)  as claimed.
Corollary 4.2 For \(n= 3\) , \(d\ge 2\)  and \(\alpha >0\) , the only non-collinear central configuration is the equilateral configuration.

Equation (REF ) implies that, if \({{Q}}_{ijk} \ne {0}\) , then the configuration is collinear (since (REF ) implies there exist three real numbers \(c_{12}\) , \(c_{23}\) , \(c_{31}\)  such that \(c_{12}{{q}}_{12} = m_3 {{Q}}_{123}\) , \(c_{23}{{q}}_{23} = m_1 {{Q}}_{231}\) , \(c_{31}{{q}}_{31} = m_2 {{Q}}_{312}\) , and it is easy to see that \({{Q}}_{123} = {{Q}}_{231} = {{Q}}_{312}\) ). Therefore, if the configuration is not collinear, \({{Q}}_{ijk} = {0}\) , and by Lemma REF  the configuration is an equilateral triangle.
Another easy consequence of Lemma REF  is the following proposition (see [4]}, [5]}, [1]} for its importance in estimating the number non-degenerate planar central configurations of four bodies).
Corollary 4.3 
For \(n=4\) , \(d\ge 2\)  and \(\alpha >0\) , if a central configuration has three collinear bodies, then it is a collinear configuration.

Assume that \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  are collinear, and \({{q}}_4\)  is not. Then, equation (REF ) implies that for suitable real numbers \(c_{12}\) , \(c_{23}\)  and \(c_{31}\)  the following equations hold:
\(c_{12} {{q}}_{12} = m_3 {{Q}}_{123} + m_4 {{Q}}_{124}, \qquad c_{23} {{q}}_{23} = m_1 {{Q}}_{231} + m_4 {{Q}}_{234},\\c_{31} {{q}}_{31} = m_2 {{Q}}_{312} + m_4 {{Q}}_{314}.\) 
This implies that there are \(\tilde{c}_{12}\) , \(\tilde{c}_{23}\)  and \(\tilde{c}_{31}\) 
such that
\({{Q}}_{124} = \tilde{c}_{12} {{q}}_{12}, \qquad {{Q}}_{234} = \tilde{c}_{23} {{q}}_{23}, \qquad {{Q}}_{314} = \tilde{c}_{31} {{q}}_{31},\) 
and by Lemma REF  this implies that \(\vert {{{q}}_{14}}\vert =\vert {{{q}}_{24}}\vert  = \vert {{{q}}_{34}}\vert \) , which is not possible since \({{q}}_1\) , \({{q}}_2\)  and \({{q}}_3\)  are collinear.
Corollary REF  can be easily generalized to arbitrary \(n\)  as follows:
Corollary 4.4 
For \(n\ge 4\) , \(d\ge 2\)  and \(\alpha >0\) , if \(n-1\)  of the bodies in the central configuration are collinear, then all of them are.

Corollary 4.5 For \(n\ge 4\) , \(d\ge 3\)  and \(\alpha >0\) , if the first \(n-1\)  bodies \({{q}}_1, \ldots , {{q}}_{n-1}\)  in a central configuration belong to a plane \(\pi \subset E\) , and the \(n\) -th body \({{q}}_n\)  does not belong to the plane \(\pi \) , then the distance between \({{q}}_n\)  and any \({{q}}_j\)  does not depend on \(j=1,\ldots , n-1\) , i.e., there exists \(c>0\)  such that \(\vert {{{q}}_n - {{q}}_j}\vert  = c\)  for all \(j<n\) . Hence the \(n-1\)  coplanar bodies are cocircular.

For each \(i,j\le n-1\)  there exists \(c_{ij} \in \mathbb {R}\)  such that
\(c_{ij} {{q}}_{ij} = \sum _{k\notin \lbrace i,j\rbrace } m_k {{Q}}_{ijk} = \sum _{k\notin \lbrace i,j,n\rbrace } m_k {{Q}}_{ijk}+ m_n {{Q}}_{ijn}.\) 
The term \(\sum \limits _{k\notin \lbrace i,j,n\rbrace } m_k {{Q}}_{ijk}\)  is parallel to the plane \(\pi \) , while the sum \(c_{ij} {{q}}_{ij} - m_n {{Q}}_{ijn}\)  is a vector parallel to the plane containing \({{q}}_i\) , \({{q}}_j\)  and \({{q}}_n\) . Being equal, they both need to be parallel to both planes, and hence they are multiples of \({{q}}_{ij}\) . Therefore, by Lemma REF , there exists \(\tilde{c}_{ij}\in \mathbb {R}\)  such that \({{Q}}_{ijn} = \tilde{c}_{ij} {{q}}_{ij}\) , and as above this implies that \(\vert {{{q}}_{in}}\vert  = \vert {{{q}}_{jn}}\vert \) .
Pyramidal configurations for \(d=3\)  and \(\alpha =1\)  were studied in first [7]}; see also [8]} for applications to perverse solutions and for the value of the constant \(c\) .
Acknowledgements
Work partially supported by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems COMPAT”.
[1]Referencesref
